Equivalent of the old beauty button for subdivide?
You can select an Edge Ring (sort of the opposite of an Edge Loop) with Ctrl+Alt+Right Click and subdivide that.



Also remember there's Subdivide Smooth as well, which is nice for interpolating the curvature of the model (though it cannot do this on a flat 2D surface like this example).

Unfortunately it won't really help in that Icosphere case though.